Let's look at adding and subtracting fractions with the same denominators. Remember that fractions have a part-to-whole relationship. Look at the fraction two fifths and the model.

We see that 2 parts are shaded out of the whole…

…the total of 5 parts. Let's look at what happens when we add fractions with the same denominator.

Let's add the fraction one fifth to the fraction two fifths.

The circles for both fractions are divided into the same number of fair shares. What changes is the number of parts.

When we add the parts together, the whole, or denominator, stays the same.

We get 3 parts out of the total of 5 parts, or three fifths. 8

This process does not work as easily when the wholes, or denominators, are not the same. Let's look at an example. Here we see the fractions one half…

…and one third. 10

The denominators are not the same.

We can combine the parts, but we cannot tell what the answer is We can combine the parts, but we cannot tell what the answer is. Should the answer be in halves or thirds or something else? We can see what the answer looks like and could probably give an approximate answer to the problem if we had to. But, we cannot give an exact answer.

To get an exact answer when we add and subtract fractions with unlike denominators, we need to use the same fair share for both fraction. This is the common denominator. 

One way we can find a common denominator is to use fraction bars One way we can find a common denominator is to use fraction bars. Here we see the fraction bars for one half and one third.

Both halves and thirds line up with sixths. We see that one half…

…is equivalent to three sixths.

We also see that one third…

…is equivalent to two sixths.

Now that we have found our equivalent fractions…

…we can rewrite the problem using fractions that have the same fair shares.

We add our new fractions using fraction bars.

We combine the fractions to find the answer.

Three sixths…

…plus two sixths…

…equals five sixths.

It’s important to remember that when we make denominators the same, we also make equivalent fractions.

How do we make equivalent fractions?

Let’s look at two ways to find a common denominator Let’s look at two ways to find a common denominator. One way is to use a table of multiples. We’ll find a common denominator for the denominators three and seven.

We start with multiples of three. Three times one is three…

Three times two is six.

Three times three is nine.

Three times four is twelve.

We find the next multiple: three times five is fifteen.

Three times six is eighteen.

Three times seven is twenty-one.

We’ll keep going. Three times eight is twenty-four…

…and three times nine is twenty-seven.

The last multiple we’ll look at is three times ten, or thirty

Now we look at multiples of seven. Seven times one is seven Now we look at multiples of seven. Seven times one is seven. We keep going…

…until we line up with a multiple of three. Seven times two is fourteen.

We try one more, Seven times three is twenty-one.

The first multiple three and seven have in common is twenty-one The first multiple three and seven have in common is twenty-one. This means that a common denominator for the denominators three and seven is twenty-one. Using a table of multiples helps us find the least, or smallest, common denominator.

Let’s use this information to solve the problem two thirds minus one seventh.

We start with the fraction two thirds.

We want to multiply by a fraction equal to one…

…that will give us a denominator of twenty-one …that will give us a denominator of twenty-one. We think, three times what number is twenty-one?

The answer is seven, so we multiply by seven sevenths, or one.

We see that two thirds is equivalent to fourteen twenty-firsts.

Now let’s look at the fraction one seventh.

Again, we multiply by a fraction equal to one…

…that gives us a denominator of twenty-one …that gives us a denominator of twenty-one. What number times seven is twenty-one?

The answer is three. We multiply by three thirds, or one…

…and get three twenty-firsts. This is equivalent to one seventh.

Now we rewrite the problem using our equivalent fractions Now we rewrite the problem using our equivalent fractions. Fourteen twenty-firsts minus three twenty-firsts…

…equals eleven twenty-firsts.

We can check our work using fraction bars.

We start with the fraction bar for fourteen twenty-firsts…

…and take away three twenty-firsts…

…and see that we are left with eleven twenty-firsts …and see that we are left with eleven twenty-firsts. Our answer is the same.

Now let’s look at another method for finding common denominators Now let’s look at another method for finding common denominators. We can multiply the denominators together.

Let’s look at the problem seven tenths minus three twentieths.

One way to get a common denominator for both fractions is to multiply the denominators. Ten times twenty…

…is two hundred.

This means that a common denominator for these fractions is two hundred. This is not always the least common denominator, but it is a common denominator.

Now we find equivalent fractions to complete the problem Now we find equivalent fractions to complete the problem. We want to multiply seven tenths…

…by a fraction equal to one…

…that will give us a denominator of two hundred.

We know the fraction is twenty twentieths…

…so the equivalent fraction is one hundred forty, two hundredths.

Now we’ll find an equivalent fraction for three twentieths.

We multiply by a fraction equal to one…

…that will give us a denominator of two hundred.

We know we need to multiply by ten tenths…

…and we get thirty two hundredths.

We rewrite the problem using the new fractions We rewrite the problem using the new fractions. One hundred forty, two hundredths minus thirty two hundredths…

…is equivalent to the original problem.

The difference is one hundred ten two hundredths.

What are some common situations that require addition and subtraction of fractions?

There are times when we need to add or subtract fractions to solve a problem, like this one. Hector rides his bicycle to school every day.

He rides two and one third miles down Perry Street to Grove Street.

From grove street he rides one fourth of a mile to the school.

How far does Hector ride to school?

We want to know how far Hector rides to school.

To solve the problem, we need to add two and one third and one fourth.

The denominators are not the same, so we have different fair shares The denominators are not the same, so we have different fair shares. We need to find a common denominator for the two fractions.

We’ll use our steps for adding fractions with unlike denominators.

First, we want to find the least common denominator, or LCD First, we want to find the least common denominator, or LCD. We’ll use a table of multiples to help.

We see that the first, or least, common multiple of three and four is twelve.

Now we can find equivalent fractions by multiplying by a fraction equal to one that will give us a denominator of twelve.

We start with one third.

We multiply by four fourths…

…and get four twelfths.

Now we look at one fourth.

Here we multiply by three thirds…

…to get three twelfths.

Now we have fractions with the same fair shares.

We can add the numbers together. Don’t forget the whole number We can add the numbers together. Don’t forget the whole number. Two and four twelfths…

…plus three twelfths…

…equals two and seven twelfths.

Hector rides his bike two and seven twelfths miles to school.